Weekly summary 2017-04-29

Day by day

• Mon. 4-24: If I can bound \[ \an{\mathcal L g, \an{\nb f, \nb g}} \ll \an{\mathcal L g, \mathcal L g} \] independent of \(f\), then I can argue that for small \(\de\), eigenvectors for Langevin on \((1-\de)f\) are close to eigenvectors for Langevin on \(f\). (One has to be careful with which eigenspaces to work with.)

  I don’t know how to do this. The best I can do is \[\begin{align} \an{f, \mathcal L_\mu f}_\mu &\le -k \ve{f}_\mu^2\\ \implies \an{f, \mathcal L_{\mu'} f}_{\mu'} &\le -\fc{k}{1+O(\de)}\ve{f}_{\mu'}^2. \end{align}\] where \(\mu'\) is for \((1-\de)f\), which gives an angle between eigenspaces of \(1+O\pf{\la_k}{\la_l}\) where \(\la_k,\la_l\) are the threshold values for the eigenspaces. This does NOT go to 1. I need something that goes to 1.
• Tue. 4-25:
  • Perceptron writeup.
  • Experiment with semi-random features.
    • Distribution is normal, square loss. (But maybe square loss is just a proxy?)
    • Doesn’t seem to work: loss on batch of 100 is on order of 20, even with 50 nodes.
  • Can estimate gaussian robustly given samples from something with \(L^\iy\) distance (in log space) \(h\) with \(\wt O (h^2)\) samples. Can we extend this to convex things?